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For a general sequence of objects and morphisms, we construct two N-complexes. Then we can define cohomologies (i, k)-type of the N-complexes not only on a diagonal region but also in the triangular region. We obtain an invariant defined on a general sequence of objects and morphisms. For a short exact sequence of N-complexes, we get the associated long exact sequence generalizing the classical long exact sequence. Lastly, several properties of the vanishing cohomologies of N-complexes are given.
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Naoya Hiramatsu, Goro C. Kato, Urcohomologies and cohomologies of <var>N</var>-complexes. Port. Math. 67 (2010), no. 4, pp. 511–524